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      The present value of a number of cash flows discounted by the same yield will be

1.13

Comparing (1.13) with (1.5), we find that they are identical if we make the following substitutions:

       1.14

      We can derive the continuously compounded yield and durations in the limit as

:

      1.15

      In the continuously compounded framework, duration (D) and convexity (X) become much simpler to handle, and modified duration and Macaulay duration converge to the same value:

      1.16

      1.17

      The change in the price of a security due to a small change in its yield in the continuously compounded framework is

       1.18

      1.3 PORTFOLIO ANALYTICS

      A bond portfolio can be composed of many bonds along the maturity, credit quality, and currency spectrums. For regulatory, policy, or strategy purposes, the portfolio manager needs to know the duration of the portfolio. Since different market sectors may have different coupon frequencies, it is important that all calculations for the duration be done on a consistent basis.

      Most bond portfolios are managed against a benchmark. The benchmark can be an index or it can be the peer group. In the cases of indices, such as the Barclays Aggregate Bond Index, the composition of the index is known on or before the last business day of a month for the following month. Portfolio managers can adjust the duration of the portfolio in relation to the changes in the duration of the index. A benchmark can be a peer group where the duration of the benchmark cannot be measured, but can be estimated through market movements. We will cover this issue in more detail in Chapter 19.

      Before we calculate the duration of a portfolio, we introduce the concept of the value of a basis point (VBP) or dollar value of a basis point (DV01), which is the change in the market value of a portfolio resulting from the change of 1 basis point in the level of interest rates:

      1.19

      where M_j and D_j are the market value and duration of a bond j in the portfolio. Consider a portfolio of N securities, each with multiple cash flows. The total market value of the portfolio can be written as

      1.20

      where p_j is the price of security j, w_j is the weight or face amount of security j, c_ij is the ith cash flow of security j and t_ij is the time to that cash flow. For a small uniform change in the yield of all bonds in the portfolio, it can easily be shown that the change in the market value will be

      1.21

      Alternatively,

      1.22

      If D is the duration of the portfolio, the change in market value for a change of

in yield will be

      1.23

      The duration of the portfolio is

      1.24

      Thus, the duration of a portfolio is the market value weighted sum of the duration of all bonds in the portfolio. Alternatively, the duration of a portfolio is the sum of all the VBPs divided by the market value and multiplied by 10, 000.

      To estimate the yield of the portfolio, we note that the market value of the portfolio can be calculated by discounting all bonds in the portfolio by their respective yields or by discounting all the cash flows in the portfolio by the portfolio yield:

      1.25

      where y is the overall yield of the portfolio and y_j is the yield of bond j. Subtracting the summations and expanding the resulting difference by Taylor series and retaining only the first two components leads to

      1.26

      The portfolio yield can now be calculated:

1.27

The yield of a portfolio as calculated by (1.27) can be significantly different from the market value weighted yield in a non-flat yield curve environment.

      The conventional duration of a portfolio also requires some adjustments in a non-flat yield curve environment. If M and D are the market value and duration of a portfolio respectively, and D_j and X_i are the duration and convexity of a security, then

      1.28

      Expanding the summation using Taylor series and keeping only the first two components leads to

1.29

For a portfolio of two zero coupon bonds, (1.29) simplifies to

1.30

Table 1.1 shows an example of the yield and duration of a portfolio of two zero coupon bonds. The market value weighted yield of the portfolio is 2.5  % (line 3) which is significantly below the actual yield (3.714  %). The actual yield is calculated by iteratively finding the yield that correctly reproduces the market value of the portfolio. The duration weighted yield on line 4 (3.7  %) is very close to the actual yield of the portfolio in line 5.

Table 1.1 Yield and duration of a portfolio

The correct method for calculating the duration of A + B as one security is by discounting each cash flow by its respective discount yield and then summing the contributions from all cash flows, resulting in a duration of 10 years. Calculating the duration of the combined securities by discounting all the cash flows by a yield of 3.714  % on line 5 will result in a duration measurement that is off by about 0.42 years from the correct duration. The weighted duration calculated from equation (1.30), shown on line 4, is very close to the market convention duration of the combined securities.

      When we combine two securities into one, in a steep yield curve environment,

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